The 3-Space PG(3,2) and Its Group

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THE 3-SPACE PC?(3, 2) AND ITS GROUP.*
By George M. Conwell

Introduction. An n-space in which the points are determined by
homogeneous coordinates will contain only a finite number of points if the
coordinates are restricted to be integers reduced to a modulus. In this paper
the modulus 2 is used, i. e., the only numbers possible are and 1.

The configuration of the "real" points, i. e., those whose coordinates are
expressed in terms of or 1, is studied in detail.

The complete projective group of coUineations and dualities of the
3-space is shown to be of order |8 and to have as a sub-group the linear

15 ~

homogeneous group L. H. G. |8, this sub-group consisting of the projective

coUineations of the 3-space. The line coordinates for the lines of the 3-space
are introduced and the linear complexes and congruences obtained. These
line coordinates can also be considered as the coordinates of points in a
5-space. To every transformation of the 3-space there corresponds a trans-
formation of the 5-space. In the 5-space, there are determined 8 sets of 7
points each, "heptads," by means of which is established the isomorphism

IB

of the linear homogeneous group L. H. G. 18 with the alternating group

on 8 letters.

An 8 letter notation is derived from the "heptads" for the points, lines,
and planes of the 3-space.

The geometry gives a complete solution of Kirkman's School Girl
Problem and is related to several functions which are of importance in the
Galois theoiy of equations. The configuration of the 3-8pace is the same as
that studied by Moore in this connection. t

The Configuration of the " Real " Points. A point is determined
in a 3-space by 4 homogeneous coordinates (a, b, c, d) and when the numbers

♦Oswald Veblen and W. H. Bussey, Finite Projective Geometries. Transactions of
the American Mathematical Society, \ol. 1 (1906), pp. 241-259.

t E. H. Moore, Concerning tlie General Equations of the Seventh and the Eighth Degrees
Mathematische Annalen, vol. 51 (1899), pp. 417-444,
(60)

THE 3-SPACE PG(3,2) AND ITS GROUP 61

a, b, c, d, are integers and reduced modulo 2 we obtain 15 points excluding the
combination (0, 0, 0, 0) ; these are called the "real" points of the 3-space.
If Pa, = («n ^1. Cv <^i) and -/'., = («3» ht ^2, d^) are any two points, the
points of the line joining them are given by

the only possible sets of values for (\, 17) are (1, 0), (0, 1), (1, 1), hence
the number of points on a line is 3. The points of a line may thus be denoted
by P„^, P^, Pa, + a,- Consider a fourth point P„., which is not on the line
-P-H. Pa,, Pa, + a,l evldoutly Pa,+a-y Pa^+a', -Pa,+.,+.', aro poiuts of tho llncs
joining P„. to the points of the line P.^, P,^, Pa,+a^-

These 7 points are all the points of a plane, since any 2 points P^ and
Pj, determine a collinear point Px+„, which is contained among the 7. The
configuration of the 7 points of a plane is that of a complete quadrangle in
which the diagonal points are collinear. The number of lines in a plane is 7
as there are 7 points and 3 lines through each point.

In the 3-space the number of "real" planes is the same as the number of
"real" points, since for each possible set of point coordinates there is a possi-
ble set of plane coordinates. Consider the 7 points of a plane which can be
denoted by

and a point Pa" not upon the plane ; this will determine 7 other points

p p p p p p p

-*■ a^ + a*'5-* dj-J- a '> -*■ Oi + aj + «"» -*■ a' -{- a"l -*■ aj + a' + a"» -* 02 + a' -f"*""' «i

+ <«j + «'+ «"■

These 15 points constitute all the "real" points of the 3-space, for any 2
points Pa, and Pj, determine a collinear point Px+y, which is included among
the 15 and any 3 non-collinear points Pj,, Py, P^ determine the 7 points of a
plane, which can be denoted by

p p p p p p p »

■^ xf -^ y -^ -+»' ■'■ zf -'■ x + *' "'■ y + zf '■ x+y+s-

Any 4 non-coplanar points determine 15 points. This process of
"doubling" from a point without an w-space always leads to an (n + 1) -space
or from a PC?(n, 2) to a P(^(n + 1,2).

Through each point there are 7 lines, for take any point and a plane not

♦ E. H.lMoore, loc. cit.

62 CONWELL [January

containing it, the 7 lines joining this point to the points of a plane contain all
l-*) points. The number of lines is 35, since there are 15 points and 7 lines
through each point and 3 points on a line : 7.15/3 = 35.

Of the 15 planes 3 pass through each line. Consider a plane and a point
outsid3 ths plane, the point with the 7 lines of the plane determines 7 dis-
tinct planes, one of which contains any 2 points of the 3-space, hence there
are 7 planes through each point.

The whole configuration can be exhibited in the table *

s.

Si

S,

^0

15

7

7

^1

3

35

3

s.

7

7

15

So is a point, /6\ a line, S-i a plane and in general S„ is an «-space. A
number n in the ith row and the ^th column gives the number of i-spaces in
the given space, while a number n in the ith column and the A;th row gives the
number of i-spaces which are united with a A;-space.
By the use of the transformation T of period 15

Xt-Xi+ + Xi Xi = x[ + x's + x'i

Xj = Xj + Xg + X4 x^ = x[ + x'i

T or

xi = Xi + X^ + X3 X.i= x'i + x[

xi = Xj + a^ + a-g + 0:4 X4 = + x'3 + x[

all the points can be obtained from a single one, say (1, 1, 1, 1), and all the
planes from a single one, say Xj + X2 + X3 + x^ = 0.

1 (1,1,1,1) I X1 + X2 + X3-I- X4=0

2 (0,1,1,0) II X4=0

3 (0,1,0,0) III X3 4- 0:4 =

* E. H. Moore, Tactical Memoranda I-III. American Journal of Mathematics, vol. 18
(1896), pp. 264-303.

1910] THE 3-8PACB PG (3,2) AND ITS GROUP 63

Hence denoting the points by the numbers 1 to 15, the point i is ob-
tained from the point 1 (1,1,1,1) by the transformation T'. All the planes
are obtained from one expressed through its 7 points as 1 (1,2,5,7,12,13,14)
by the transformation T (1,2,3,4, • • •, 14,15) and its powers.

And all the lines can be obtained from 3, each expressed in terms of its
3 points, (1, 2, 5), (1, 3, 9), (1, 6, 11), by means of the same transforma-
tion T. The first two lines have each 15 conjugates while the third has 5
under the transformation T and its powers.

Introduction of Line Coordinates. Two points Pa and Pa' of the
3-space determine a line: P. (a^, a.^, a^, a^) and P^'(«(, al, a^, «i).
There are 12 determinants j3,j of the form

Pik =

a,: a.

which are related two by two, pi, = p„i (since ^ = — P modulo 2) .

The 6 quantities {Pm^ Pis, Pu, Pis^ Pn^ Pzi) ^^^ the Plucker line coordi-
nates and completely determine a line. Thej' are related by the condition

PnPu +P^Pn +PuP'is = 0.*

The line coordinates of the 35 lines can be obtained from those of the 3
lines (1, 2, 5), (1, 3, 9), (1, 6, 11) by means of a transformation P on the
p's which is induced by the transformation T:

Pii=Pii +Pn

p'w=Pii+Piz+Pn +PU + PU:

P'u = Pli + i>13 + PU + Pn

P^ = +Pn + PU + JP23 + Pn + Pu

P'u= +Pn +Pi3 +Pu

P'u= Pu-\- +Pi4.+Pzt.

The 3 lines and their line coordinates are respectively (1, 2, 5),
(1, 3, 9), (1, 6, 11), and (1,1,0,0,1,1), (1,0,0,1,1,0), (0,1,0,1,0,1).

It is evident that the line coordinates may be looked upon as points in a

•C. M. Jesaop, Treatise on the Line Complex, page 17.

64 CONWELL [January

5-space, a PG{b, 2). In such a space there are 2® — 1 = 63 points excluding
(0,0,0,0,0,0). Of these 63 points, the 35 representing lines of the 3-space
lie upon a surface, 8,

PnPu + PkP'U + PiiPia = 0-

The lines whose coordinates satisfy a linear equation constitute a linear
complex, which can be either degenerate or non-degenerate.

A degenerate complex consists of all the lines which meet a given line,
called the axis, together with this axis. Given two lines p and p' with the
line coordinates {Pu, Pn, Pu, Vn^ Pa-, Pu) and {p'a, Pi3,Pu, p'^, p'n^p'u)^ the
condition that they may intersect is

PnP'u+PviP'n +PuPk +P2zP'u+Pit.P'iz +PuPvi = 0.

Hence this condition, if them's are considered constant, is the equation
of a degenerate complex, consisting of all lines meeting p' and of p' itself.
The equation is satisfied for^' since the equation then reduces to the condi-
tion jS.

Any linear relation among the jj's, as

apii + bpu + cpu + dp^ + ePu + fPu = 0,

can be looked upon as the polar of the point (f,e,d,c,b,a) with respect to the
surface S and if this point in the 5-space represents a line in the 3-space, then
the complex determined will be degenerate ; if it does not represent a line,
the complex is non-degenerate. Hence the polar of a point on the surface iS,
i. e., the tangent 4-space at the point to the surface S, has in common with the
surface S the points representing the lines of a degenerate complex, while the
polar of a point off the surface S determines in a similar way a non-degen-
erate complex. The number of degenerate complexes is equal to the number
of points in the 5-space which represent lines in the 3-space, that is 35.
They may all be obtained from the following three by the transformation P
and its powers :

Pl2 + Pl3 + Pu + Pu = 0, Pi2 + i523 + P34 = 0, P13 + P^ + Pu = 0.

A non-degenerate complex is a system of lines such that through every
point of the 3-space there is a flat pencil of lines and all the lines in every
plane pass through a point. The proof is the same as in the ordinary case.*

♦ Jessop, loc. cit., page 25.

1910] THE 3-SPACE PG (3, 2) ASD ITS GROUP 65

The lines whose coordinates satisfy two linear equations and hence are
common to two complexes, constitute a congruence. Consider the two com-
plexes G and C" :

apii + 6pi3 + <^I>u + dp^ + «Pu +fPu = 0,
a'Pn + ^'Pii + '^'Pu + d'p^ + e'pn -\- fpu = ;

these determine a third complex, G" = C + G\

(a + a')iJi2 +(6 + ftOi'is H-Cc + "OiJu +((^ + (^') Pas +(« + ^')P^

G, G' and G" are degenerate if the points {a,b,c,d,e,f), (a',b',c',d',e',f') and
(a + a',b + b\ c + c',d + d\e + e', f + /' ) respectively are upon the surface
S. The condition that this last point is on the surface reduces to

af + a/+ 6e' + Ve + cd' + c'd = 0.

This is the condition that the axes of G and G' intersect. Hence if the
three complexes of a family are all degenerate, the congruence determined
consists of 11 lines meeting the two intei'secting axes of the complexes G and
G' . If two of the complexes of the family are degenerate, while the third is
non-degenerate, the congruence consists of 9 lines meeting^ the two non-inter-
secting axes of the degenerate complexes. If only one of the complexes of
the family is degenerate, then the non-degenerate complexes contain the axis
of the degenerate complex, the congruence consists of 7 lines, the axis
and 6 lines meeting it, one line through each point of the 3-space. This is a
degenerate congruence with an axis. When the three complexes of a family
are all non-degenerate, there is a non-degenerate congruence determined, which
consists of 5 lines such that there is one and but one through each point of
the 3-space.

The Group of the 3-space. Every projective collineation* in the
3-spac!) PG(3, 2) is represented by a transformation T on the point
coordinates

« = 4

T: x'^= '^a^iX, r= (1, 2,3, 4).
♦ O. Veblen and W. H. Bussey, loc. cit., p. 253.

66 CONWELL [January

This transformation is completely determined if we know into what
points the vertices of the tetrahedron of reference are transformed ; (1,0,0,0)
goes into (a^, a^, CTsi, a^), (0,1,0,0) goes into (a^^, a^, a^, ff«), etc.

In order that the transformation may have an inverse, the determinant of
the transformation must be different from zero, and this is the condition that
the four points {a^, a^f, a^i, a^j), (i = 1,2,3,4) into which the vertices of the
tetrahedron are transformed shall not be coplanar. Hence the number of
transformations of the group of projective collineations is equal to the number
of ways four non-coplanar points can be chosen, i. e., 15-14-12-8 = 20,160
= |8/2. The dualities or polarities of the 3-space are transformations of the
form

i = 4

Xr=^^ariU, r=(l, 2, 3, 4),

where the a;'s are point coordinates and the m's plane coordinates.

These transformations have the property of changing points into planes
and vice versa, but change lines into lines. The number of these dualities is
evidently the same as the number of projective collineations 18/2. The order
of the complete projective gtoup, consisting of all projective collineations
and dualities, is 18.

Since a transformation of the complete projective group changes lines
into lines, every transformation of this group determines a transformation on
the line coordinates. Conversely every transformation on the line coordinates
which changes lines into lines corresponds to a transformation of the complete
projective group. Let the transformation on them's be of the form

i = 6

pj.=2«rii'i' r = (l, 2, 3, 4, 5, 6),

t'sl

The lines with the coordinates (1,0,0,0,0,0), (0,1,0,0,0,0),

(0,0,1,0,0,0), (0,1,1,0,0,0), (1,0,1,0,0,0) and (1,1,0,0,0,0), all pass
through the point (1,0,0,0) and must be transformed into lines, hence we
must have

(A) aijCiei + a^iOM + a3,-a" = 0, (t = 1, 2, 3),

(B) (au+ ay) (a^i + a^)+(aii + a^) {a^ + af^)■k■{a^^ + a^) (a« + a^)= 0,

1910]

THE 8.SPACE PQ (S, 2) AND ITS GBOUP

67

(5) reduces by means of (^) to {B') :
{B') ttuU^ + a^a^ + a^^^ + a^a^ + a^^^^ + a^Uii = 0, {ij = 1, 2, 3).

(B') is the condition that the three lines (1,0,0,0,0,0), (0,1,0,0,0,0),
(0,0,1,0,0,0), which are not in a plane, must be transformed into lines which
meet two by two, hence they must pass through a point or lie in a plane.
Hence all the lines through a point must be transformed into lines through a
point, a projective coUineation, or be transformed into the lines of a plane, a
duality. The simplest transformation on the p's which is a duality is

i'l=i'«. i'a=i'6. I>3=Pi> Pi=JPz^ i>5=i>j. P6=Pv

The Configuration of the 28 Points not upon the Surface S
in the 5-space. Interpreting the results of section 2 in terms of the
5-space we see that taking the polar of a point in the 5-space with respect to
the surface S determines a non-degenerate complex for the 3-space. Taking
the polars of tliree points of a line in the 5-space determines a non-degener-
ate congruence for the 3-space. Hence the number of non-degenerate con-
gruences is the same as the number of lines in the 5-space, which are wholly
off the surface S. Whatever is true for a single point of the 28 is true for
all, as any point can be transformed into any other. The number of lines
wholly off the surface S which can be drawn through the point (1,0,0,0,0,1)
is 6 and the same number can be drawn through each of the 28 points, hence
there are 28-6/3 = 56 lines wholly off the surface S. The number of non-
degenerate congruences is thus 56.

Consider the 3 points of a line as J.= (1,0,0,0,0,1), B = (1,0,1,1,0,0),
0= (0,0,1,1,0,1), which are wholly off the surface S. Through each of
these points there are 5 other lines wholly off the surface S. These lines are
denoted below in terms of their 3 points, the 3 points in a row are the 3 points
of a line.

Lines throagh A

(l,OiO,0,0,l)(l,0,l,l,0,0)(0,0,l,l,0,l)
(1,0,0,0,0,1) (0,1,1,1,0,1) (1,1,1,1,0,0)
(1,0,0,0,0,1) (1,1,0,0,1,0) (0,1,0,0,1,1)
(1,0,0,0,0,1)(0,0,1,1,1,1)(1,0,1,1,1,0)
(1,0,0,0,0,1) (1,1,1,0,1,0) (0,1,1,0,1,1)
(1,0,0,0,0,1) (0,1,0,1,1,1) (1,1,0,1,1,0)

Lines throngb B
(1,0,0,0,0,1) (1,0,1,1,0,0) (0,0,1,1,0,1 )
(0,1 ,1,1,0,1) ( 1 ,0,1,1,0,0) ( 1 ,1 ,0,0,0,1 )
(0,1,0,0,1,1)(1,0,1,1,0,0)(1.1,1,1,1,1)
(0,1,1,1,1,1)(1,«,1,1,0,0)(1,0,0,0,1,1)
(0,1,0,1,1,0)(1,0,1,1,0,0)(1,1,1,0,1,0)
(1 ,1,0,1,1,0) (1 ,0,1,1,0,0) (0,1,1,0,1,0)

Lines throagh C
(1,0,0,0,0,1)(1,0,1,1,0,0)(0,0,1,1,0,1)
(1,1,1,1,0,0) (1,1,0,0,0,1) (0,0,1,1 ,0,1)
(1,1,1,1,1,1)(1,1,0,0,1,0)(0,0,1,1,0,1)
(1,0,0,0,1, 1)(1,0,1,1,1,0)(0,0,1,1,0,1)

(o,i,o,i,i,o);o,i ,1,0,1,1 )(o,o,i,i,o,i )

(0,1,1,0,1,0)(0,!,0,1,1,1)(0,0,1,1,0,1)

68 CONWELL [January

It will be observed that when two points as A and B of the line ABO
are chosen, that each of the 5 other lines through A is met by a single line ot
the 5 other lines through B. The 5 points determined by the intersection of
these pairs of lines together with the 2 points A and B we designate a
"heptad." It wiU be found that the 7-6/2 = 21 lines joining these 7 points by
pairs are wholly off the surface /S. If any 2 points of the heptad are chosen
and these points are used in the same way as A and B for the deter-
mination of a heptad, the same heptad will be determined. Each of the pairs
of points AB, BO, OA determines a heptad so that each point of the 28
points off the surface /S" belongs to 2 heptads. The number of heptads is thus
28-2/7 = 8. No 6 of the points of a heptad are in the same 4-space, no 5 in
the same 3-space, no 4 in the same 2-space, no 3 in the same 1-space.

If we take the generators of the group of projective coUineations of the

15

3-space, the L.H.G 18, the matrices below are the matrices of the 6 gen

Y
erators of the group.*

B,

^2

J?8

B^

-£"«

^6

1110

0101

Olio

1110

0101

0011

0111

Olio

0011

0100

0011

Olio

1111

0010

1011

0010

1111

0010

1011

1110

1111

0101

1011

1001

From these can be calculated the corresponding transformations in terms of the
j>'s. Let the 8 heptads be given the numbers 1 to 8 ; each heptad is deter-
mined by two of its points.

(1,0,1,1,0,0) and (1,1,1,1,1,1) determine heptad 1
(0,0,1,1,0^0) and (1,1,1,0,1,0) determine heptad 2
(1,1,0,0,1,0) and (0,1,0,1,1,1) determine heptad 3
(0,0,1,1,1,1) and (0,1,1,1,0,0) determine heptad 4
(1,0,1,0,1,1) and (0,1,1,1,0,1) determine heptad 5
(1,0,0,0,0,1) and (1,0,1,1,0,0) determine heptad 6
(1,1,1,0,0,1) and (0,0,1,1,0,0) determine heptad 7
(0,0,1,1,1,0) and (1,1,0,0,1,0) determine heptad 8.
*E. H. Moore, loc. cit., page 435.

1910] THE 3-SPACE Pff (3, 2) AND ITS GEOtTP 69

The transformations in terms of the ^'s applied to the 28 points off the
surface S permute the heptads among themselves. To the 6 generators
above correspond the permutations on the heptads,

E\= (4,7)(7,6),^2=(4,7)(6,8),^3=(*,7)(8,5),
^,= (4,7) (1,5), ^5 =(4,7) (1,2), ^6 =(4,7) (2,3);

and these are the generators of the alternating group on 8 letters.

IS

Hence the I/.H. G 18 and the alternating group on 8 letters are
T
isomorphic*

A transformation on the ^'s which is a duality is equivalent to a transfor-
mation of the symmetric group on 8 letters, for example, the duality of page
67 is equivalent £o the transformation (2,7) (3,6) (4,5) on the heptads.
To the fact that by the addition of one duality to the group of projective
collineations the complete projective group can be generated, corresponds the
fact that by the addition of one uneven transformation to the alternating group
the symmetric group is generated. The complete projective group of the
3-space, P(r(3,2), is isomorphic with the symmetric group on 8 letters.

By means of the heptads, it is possible to assign an 8 letter notation to
the points of the 5-space which are off the surface S, and to study their con-
figuration. We have pointed out the fact that each point off the surface 8
belongs to 2 heptads. The notation for a point can thus be taken as {a,b),
a :^ b, and there are 8-7/2 = 28 combinations of this form corresponding to
the 28 points off the surface S. The notation for the points of a line off the
surface S is (ab) (be) (ca) , a, b, c distinct ; for the points of a line off the sur-
face ,S' by pairs determine three heptads. Thus the two letter notations for the
points must by pairs have a letter in common. There are thus 8-7-6/1-2-3 = 56
lines off the surface /S. The maximum number of points which a plane can
have off the surface /3 is six. Such a plane in teniis of its points has the no-
tation (ab) (be) (ca) (ad) (bd) (cd). A line off the surface S has the notation
(ab)(bc) (ca)- For any point off the surface S has the notation (xy) and in
order that the line deteimined by this point with (ab) may have its third point
off the surface S, x or y must equal a or b since (ab) and (xy) must belong to
a common heptad. Let x = a then y = d, di^ai^bi^c. Then the two other

♦First proved by Jordan, Traite des substitutions. No. 516. E. H. Moore has given a
proof based on tlie same system of generators, loc. cit., page 432.

70 COKWELL [January

points of the plane which are off the surface S are (bd) and (cd) . There
are 8-7-6-5/1-2.3-4: = 70 planes of this kind.

In a similar manner the 3-spaces and the 4-8paces with the maximum
number of points off the surface S can be determined. The configuration of
the w-spaces with the maximum number of points off the surface S is given
below in a table whose interpretation is the same as that of the table page 62

So

S,

S,

^3

S,

>So

28

6

15

20

15

^1

3

56

5

10

10

s.

6

4

70

4

6

'Ss

10

10

5

56

3

'S,

15

20

15

6

28

The Conflguration of the Thirty -Five Points upon the
Surface S in the 5-8pace. The surface S has as its equation

PiiPsi + PiaPii + PuPis =

and is satisfied by the 35 points of the 5-space representing lines in the
3-space. Consider any two points upon the surface S, (ctj, Jj, Cj, (Z,, ei,fi) and
(a.j, b.i, c-i, di, e.i,fi) the third point of the line, (a^ + a^, b^ + b^, q + Cj,
di + d^, Ci + eg, fi + fi) will be upon the surface S if

(«i + «i) (/i + /2) + (*i + *-2) (ei + 62)+ (Ci + C2) (di + d,)=0
or if ai/2 + ttifi + SjCg + b-iCi + c^d.^, + c^di = 0.

This latter is the condition that the two lines in the 3-spacc represented
by the first two points intersect. Any two points of a line can be taken as
the first two, hence the 3 lines in the 3-space which are represented by the
points of a line in the 5-space must meet by pairs in the 3-space and thus lie
in a plane, or pass through a point.

Let A = (a„ a.^, a^, a^), B = {bi, b^, 63, 64), (7= (ci, c^, Cg, C4) be three
points not on a line in the 3-space. The line coordinates of AB and AC
are

(ajAj + Oj^i, • • • , aj}^ -I- 0^63) and {a^c^ + OjCj, • • • , a^c^ + a^Cg) .

1910] THE 3-SPACE PG (3,2) AND ITS GEOUP 71

These line coordinates considered as coordinates of two points in the 5-space
determine a collinear point,

(o [pi + Ci] + a.i[bi + ci], . . ., a^lbi + c^] + Uilbs + Cg])

whose coordinates are the line coordinates of the line joining A to the third
point of BC, {bi + Cj, b^ + Cj, 63 + C3, b^ + c^). Hence the three points of a
line in the 5-space which is entirely on the surface S represent a flat pencil,
three lines in the 3-space which lie in a plane and pass through a point.

The seven lines in a plane of the 3-space are represented by the 7 points
of a plane wholly on the surface xSinthe 5-space, since in the plane in the 3-space
every line meets every other line. In a similar manner the lines through a
point in the 3-space are represented by the points of a plane which is wholly
on the surface S in the 5-space.

The tangent 4-space at a point of the surface S has already been shown
to have 19 points in common with the surface S, representing the 18 lines
which meet a given line in the 3-space, and the line itself.

Since through a given line in the 3-space there are 3 planes and through
each point in a plane 3 lines, it follows that the 19 points which the tangent
4-space has in common with the surface S are upon 9 lines through the point
of tangency.

Any 4-space passing through the point of tangency has in common with
the tangent 4-spaco a tangent 3-space. These tangent 3-spaces are of two
kinds. The first kind has in common with the surface 3 II points, which
are upon 5 lines through the point of tangency, representing a degenerate
congruence determined by two degenerate complexes whose axes have a point
in common. The second kind has in common with the surface S 1 points,
which are upon 3 lines through the point of tangency ; these represent the 7
lines of a degenerate congruence with an axis.

The tangent 2-spaces are determined by two 4-spaces through the point of
tangency and by the tangent 4-space. They are of 5 kinds. One and two
have 7 points in common with the surface S and are on 3 lines through the
point of tangency. They represent in the 3-space the lines of a plane and
the lines through a point respectively. Three has 5 points in common with
the surface iS, which are on 2 lines through the point of tangency and repre-
sent in the 3-space a line and a flat pencil through each of two points of the
line determining two planes through the line. Four has three points of a line

72 CONWELL [January

in common with the surface S and represents a flat pencil of three lines in the
3-snace. Five is a tangent plane which has but a single point in common
with the surface /S and represents a single line in the 3-space.

In the configuration of the 28 points off the surface S there are 70 planes,
which have 6 points off the surface S. Two of these planes pass through
each of the 35 points of the surface 8. We have seen that the notation for
the points of one of these planes is (ab) (ac) (cb) {ad) (bd) {cd) and the nota-
tion for tlie plane itself may be taken as {abed) .

If the heptads are denoted bj the 8 letters a,b,<:,d,e^,g,h, it will be
found that the 2 planes {abed) {efgh) have but 1 point in common with the
surface aS* and that these common points are the same. Ilence the points of
the surface 8 can be denoted by a double 4 letter notation, which will also be
the notation for the lines of the 3-space. On page 63 the line coordinates of
all the lines are obtained from those of the 3 lines (1,2,5), (1,3,9), (1,6,11),
by use of the transformation Tand its powers. The 8 letter notation for
all the lines of the 3-space can be obtained from the notations for the 3 lines
above, which are respectively (1278 + 3456), (1357 + 2468), (1467 + 2358),
by use of the transformation on the heptads, which is isomorphic with T, i.e.,
(17654) (283).

The Determination of an Eight Letter Notation for the Points
and Planes of the 3-space. A point of the 3-space is determined by the
7 lines which pass through it, and these correspond to the 7 points of a plane
in the 5-space.

A plane of the 3-space is determined also by the 7 lines which lie in it,
and these correspond in the 5-space to the 7 points of a plane. To each of
the points of the 5-space there belongs a double four letter notation. Hence
a point of the 3-space will be denoted by 14 sets of 4 numbers. The eight
letter notation for the point 1 = (1,1,1,1) of the 3-space is 1278 + 1458 + 1234
+ 1357 + 1256 + 1368 + 1467, 3456 + 2367 + 5678 + 2468 + 3478 + 2457
+ 2358 and the notation for the point i is obtained from this by the I'th power
of T, where T= (17654) (283).

These 14 sets of 4 numbers constitute the planes of a finite Euclidian
geometry, where the numbers are considered as points ; there are 4 points to
a plane and 2 points to a line. This same Euclidian geometry can be obtained
from the PG{Z,2) by striking out all the points of a plane, which will cut
out 3 points from each plane. Another statement of the above is that the 14

1910] THE 3-SPACE PO (3, 2) AND ITS GEOUr 73

sets of 4 numbers constitute a quadruple system,* for when any 3 numbers
are given there is always a fourth determined.

The proof is as follows. When 3 numbers as a, b, c, are given this de-
termines a line in the 5-space ; this line determines a congruence in the 3-8pace
whose 5 lines have the notation

abed + e

es nave tne notation

efgh, abce + dfgh, abcf ■\- degh, abcg + defh, and abch + defg.

But through any point of the 3-space there is but one line belonging to a con-
gruence, hence the fourth letter belonging to abc in the notation for this line
is determined when this line is given.

The group, which leaves a plane of the 3-space or a point of the 3-space
fi.Ked, is of order 14.12-8 = 1344 and this is the order of the group of the
Euclidian geometry and of the quadruple systems on 8 letters.

There are 15 quadruple systems related to the planes of the 3-space and 15
related to the points of the 3-space. The points of the 3-space are trans-
formed into the planes and vice versa by the odd transformations of the sym-
metric group on 8 letters, hence the 2 sets of 15 quadruple systems are conju-
gate under the symmetric group, while the members of each set of 15 are
conjugate under the alternating group, which changes points into points and
planes into planes.

The 8 letter notation for the points of the 3-space enables one to calculate

15 8

the corresponding transfoiTaations of the L. H. G. |8 and the (t|8 immedi-

r T

ately. The method of passing from the linear homogeneous group to the
alternating group has already been given on page 11. To pass in the reverse
direction it is only necessary to determine into what points the vertices of the
fundamental tetrahedron are transformed. This gives a method, which does
not require the direct use of a table of corresponding transformations. t

Applications to Kirkman's School Girl Problem.t The problem
is "to arrange 15 school girls in parties of 3 for 7 consecutive day's walks,
so that no 2 girls may walk together more than once during the 7 days."

* E. H. Moore, loc. clt.

t Such a table Is given by L. E. Dickson. Matkematische Annalen, vol. 6-1 (1901;, page
664.

I See Ball, Mathematical Secreations and Problems, page 89, for numerous references to
the problem.

74 CONWELL [January

CONWELL

e of the 3-space as

5, 10, 15

1467 + 2385

1, 6, 11

1567 + 2384

2, 7, 12 or

4567 + 2381

3, 8, 13

1456 + 2387

4, 9, 14

1457 + 2386

(the latter in the 8 letter notation) is obtained by polarizing for the surface 8
with respect to the 3 points of a line in the 5-space which is wholly off the
surface S, and taking the points of the 5-space which are common to the sur-
face iS and the 3 polar 4-spaces ; these common points represent the lines of a
congruence in the 3-space. The above congruence is obtained by polarizing
with respect to the 3 points

(1,0,1,0,0,1) 28

(1,1,0,0,1,0) or 38

(0,1,1,0,1,1) 23

(the latter in the 8 letter notation). The group of a congruence is thus com-
posed of all members of the alternating group which permute 3 letters among
themselves, hence is of the order 5-3/2 = 360. A day of the schbol girl
problem is evidently a congruence.

A week's solution consists of 7 congruences which do not have a line in
common. The 5 lines of a congruence can be written in the 8 letter notation
abed + efgh, abce + dfgh, abcj + det/h, abcg + defh, and a ch -\- defg,
i. e., they are given by abcx + defghjx, x = d, e, f, g, h, and the congruence
is determined by polarizing with respect to the 3 points ab, be, ca, of a line in
the 5-space. There are 2 types of congruences having a line in common with
the above congruence, T, abdx + cefghlx, x = c, e, f, g, h, obtained by polar-
izing with respect to the points ab, bd, da, and II, abcdhjx + efgx, x= a, b,
c, d, h, obtained by polarizing with respect to the points ef, fg, ge. Hence
in order that 2 congruences shall have a line in common, the lines in the 5-space
with respect to which we polarize must either have a point in common as ab
or they must not belong to the same heptad. Hence the school girl problem
consists of finding 7 lines in the 5-space which do not intersect and such that
any 2 lines always have a heptad in common.

1910] THE 3-SFACE PG (3,2) AND ITS GROUP 75

The 8 heptads give a complete solution of the problem. We take 7 of
the heptads as (1234567) and form a P(t(2,2) with heptads as points, and
the following triads as lines :

123, 145, 167, 347, 246, 257, 356-

Each of the sets of 3 numbers determines a line in the 5-space which is
wholly off the surface 8 and no 2 of the 7 lines have a point in common
and each two have a heptad in common. These 7 lines in the 5-space thus
determine a solution of the problem.

There are 30 PG^(2,2) related to 7 letters, and since there are 8 heptads
there are 30-8 = 240 solutions of the school girl problem belonging to this
geometry. The 30 PG'(2,2) belonging to a set of 7 heptads are conjugate
under the symmetric group and there are 2 sets of 15 such that the members
of each set are conjugate among themselves under the alternating group.
Hence there are 2 sets of 120 solutions each, the solutions of each set are
permuted among themselves by the projective collineation group * and one set
is transformed into the other by a polarity.

If we apply the cyclic transformation (12345678) to the above PCr(2,2)
we obtain 8 solutions, which do not have a congruence or day in common.!
These 8 solutions embrace all 56 congruences. By a transformation of
period 15 we can obtain 120 solutions from these 8 and by means of a duality
all the 240 are obtained.

Each solution is invariant under a group of order 168, since it is a
P(?(2,2) on 7 heptads. From thisPG^(3,2) we derive 240 solutions. There
are 2|15/8 equivalent spaces which are conjugate with this space, hence there
are 240-2-|15/8 solutions of the school girl problem to be obtained from these
spaces. J. Power has shown that this is the number of possible solutions,
hence each school girl solution is related to a P(T(3,2).t

Application to the Equations of the Eighth and Lower De-
grees. A particular expression, §

V = x^x^x^x^ + XiX^aJsXg + XiX.p:2,Xi + x^x^x^x^ + XiX^ajgajg + x^x^pc^x^ + XlX^Xfpc^

-|- x^x^Xf,x^ -f- ^^z^i^^ + XjXgX^Xg ■{■ Xi^x^x^x^ + x^x^^p:^ -\- XoX^Xjar^ + x^x^x^pc^,

* E. H. Moore, loc. cit., page 441.

t E. H. Moore, loc. cit., page 443.

X J. Power, Oq the Problem of the Fifteen School Girls. Quarterly Jouriuil of Mathe-
matics, vol. 8 (1867), pp. 236-251.

§ Mathleu, Journal de matMinatiqui's pures et appliques, vol. 6 (1861), pp. 241—323.
See page 291.

76 OONWELL

which is the notation for a point or a plane in the 3-space and is invariant
under a group of substitutions of order 14:'12-8 = 1344, has been used to re-
duce the general equation of the eighth degree to a special one whose Galois
group is of order 1344.* The expression v has 15 conjugates under the alter-
nating group and hence is the root of an equation of the fifteenth degree
whose coefficients can be rationally expressed in temis of the coefficients of the
original equation and of the square root of the discriminant. On adjoining
a root V of this equation ot the fifteenth degree and adjoining the square root
of the discriminant, the general equation of eight degree reduces to a particular
one with the Galois group of order 1344.

The equation of the seventh degree may be considered by allowing one
of the heptads to be fixed. A function which plays a similar role to the one
above for the equation of the seventh degree is

V = x^x^X'j + XjCCdX^ + XgpCfjX^ + XiXyCg -|- x^x^x^ + XffCgpc^ -\- x^x^XQ.

This is the notation for a school girl solution and as has been shown is inva-
riant under a group of order 168. Thus v has 15 conjugates under the alternat-
ing group. Hence on adjoining a root of an equation of the fifteenth degree
and adjoining also the square root of the discriminant, the general equation of
the seventh degree reduces to a special equation whose Galois group is of
order 168. t

Nbw Haven, Comj.,
April, 1909.

*H. Weber, Lehrbuch der Algebra, vol. 2, page 377.
t H. Weber, loc. clt , page 540.